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arXiv · 2304.04690

Digraph Colouring and Arc-Connectivity

Pierre Aboulker·Guillaume Aubian·Pierre Charbit
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Abstract

The dichromatic number $\vec\chi(D)$ of a digraph $D$ is the minimum size of a partition of its vertices into acyclic induced subgraphs. We denote by $\lambda(D)$ the maximum local edge connectivity of a digraph $D$. Neumann-Lara proved that for every digraph $D$, $\vec\chi(D) \leq \lambda(D) + 1$. In this paper, we characterize the digraphs $D$ for which $\vec\chi(D) = \lambda(D) + 1$. This generalizes an analogue result for undirected graph proved by Stiebitz and Toft as well as the directed version of Brooks' Theorem proved by Mohar. Along the way, we introduce a generalization of Haj\'os join that gives a new way to construct families of dicritical digraphs that is of independent interest.

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Pierre Aboulker, Guillaume Aubian, Pierre Charbit. 2023-04-10. Digraph Colouring and Arc-Connectivity. https://arxiv.org/abs/2304.04690

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